dvsep_c |

Table of contents## Proceduredvsep_c ( Time derivative of separation angle ) SpiceDouble dvsep_c ( ConstSpiceDouble s1 [6], ConstSpiceDouble s2 [6] ) ## AbstractCalculate the time derivative of the separation angle between two input states, S1 and S2. ## Required_ReadingNone. ## KeywordsGEOMETRY ## Brief_I/OVARIABLE I/O DESCRIPTION -------- --- -------------------------------------------------- s1 I State vector of the first body s2 I State vector of the second body ## Detailed_Inputs1, s2 are, respectively, the state vector of the first and second target bodies as seen from the observer An implicit assumption exists that both states lie in the same reference frame with the same observer for the same epoch. If this is not the case, the numerical result has no meaning. ## Detailed_OutputThe function returns the double precision value of the time derivative of the angular separation between S1 and S2. ## ParametersNone. ## Exceptions1) If numeric overflow and underflow cases are detected, an error is signaled by a routine in the call tree of this routine. 2) If called in "RETURN" mode, the function returns 0. 3) Linear dependent position components of `s1' and `s1' constitutes a non-error exception. The function returns 0 for this case. ## FilesNone. ## ParticularsIn this discussion, the notation < V1, V2 > indicates the dot product of vectors V1 and V2. The notation V1 x V2 indicates the cross product of vectors V1 and V2. To start out, note that we need consider only unit vectors, since the angular separation of any two non-zero vectors equals the angular separation of the corresponding unit vectors. Call these vectors U1 and U2; let their velocities be V1 and V2. For unit vectors having angular separation THETA the identity || U1 x U1 || = ||U1|| * ||U2|| * sin(THETA) (1) reduces to || U1 x U2 || = sin(THETA) (2) and the identity | < U1, U2 > | = || U1 || * || U2 || * cos(THETA) (3) reduces to | < U1, U2 > | = cos(THETA) (4) Since THETA is an angular separation, THETA is in the range 0 : Pi Then letting s be +1 if cos(THETA) > 0 and -1 if cos(THETA) < 0, we have for any value of THETA other than 0 or Pi 2 1/2 cos(THETA) = s * ( 1 - sin (THETA) ) (5) or 2 1/2 < U1, U2 > = s * ( 1 - sin (THETA) ) (6) At this point, for any value of THETA other than 0 or Pi, we can differentiate both sides with respect to time (T) to obtain 2 -1/2 < U1, V2 > + < V1, U2 > = s * (1/2)(1 - sin (THETA)) * (-2) sin(THETA)*cos(THETA) * d(THETA)/dT (7a) Using equation (5), and noting that s = 1/s, we can cancel the cosine terms on the right hand side -1 < U1, V2 > + < V1, U2 > = (1/2)(cos(THETA)) * (-2) sin(THETA)*cos(THETA) * d(THETA)/dT (7b) With (7b) reducing to < U1, V2 > + < V1, U2 > = - sin(THETA) * d(THETA)/dT (8) Using equation (2) and switching sides, we obtain || U1 x U2 || * d(THETA)/dT = - < U1, V2 > - < V1, U2 > (9) or, provided U1 and U2 are linearly independent, d(THETA)/dT = ( - < U1, V2 > - < V1, U2 > ) / ||U1 x U2|| (10) Note for times when U1 and U2 have angular separation 0 or Pi radians, the derivative of angular separation with respect to time doesn't exist. (Consider the graph of angular separation with respect to time; typically the graph is roughly v-shaped at the singular points.) ## ExamplesNone. ## RestrictionsNone. ## Literature_ReferencesNone. ## Author_and_InstitutionN.J. Bachman (JPL) J. Diaz del Rio (ODC Space) E.D. Wright (JPL) ## Version-CSPICE Version 1.0.1, 13-AUG-2021 (JDR) Edited the header to comply with NAIF standard. -CSPICE Version 1.0.0, 09-MAR-2009 (EDW) (NJB) ## Index_Entriestime derivative of angular separation |

Fri Dec 31 18:41:05 2021